Mean, Median, Mode
Social Statistics
This week
• Mean
• Median
• Mode
2
How do we decide which is “best”?
• The overall goal of central tendency is to find
the single score that is most representative for
the distribution.
3
Measures of Central Tendency
• Mean: Arithmetic average
• sum of scores divided by number of scores
• most frequently used b/c it uses all scores in the set
• Median: “Middle” score, when scores are in
order
• corresponds to the 50th percentile
• appropriate for skewed/open-ended distributions, and
• distributions with undetermined scores
• Mode: Most frequently occurring (popular) score
• appropriate for nominal data
4
Mean
___
X
X


n
___
X (x bar): the mean
 X : sum of the data
n : number of the data
5
Mean
• The sample mean is the measure of central
tendency which can approximate the
population mean
• The mean is very sensitive to extreme scores
– It can put the mean in some extreme direction
– Make it less representative
– Less useful as a measure of central tendency
6
Calculate mean
Location
Number of annual customers
Lanham Park Store
2150
Williamsburg Store
1534
Downtown Store
3564
The mean or average number of shoppers in each store?
Using Excel to do that
• use your own formula
• use AVERAGE function
7
Median
• It is defined as the midpoint in a set of scores
– 50% of the scores fall above and one half fall
below.
8
Calculate median
• Odd number of data
– Rank them
– Median=middle one
– Example: 10, 9, 8, 7, 5 (median=8)
• Even number of data
– Rank them
– Median= sum of two middle data/2
– Example: 10, 9, 8, 7, 6, 5 (median=(8+7)/2=7.5)
9
Median
• The median is insensitive to extreme cases,
where the mean is not.
• To measure the central tendency:
– Have some extreme data, using median
– No extreme data, using mean
– Example: 14, 3, 2, 1, (mean=5, median=2.5)
• Which represents better the central tendency?
10
Median in Excel
• Calculate the median of income level
11
Mode
• The mode is the value that occurs most
frequently.
– Calculate the frequency of all the values in a
distribution
– The value that occurs most often is the mode
12
Calculate mode
• 185 students:
Student distribution
Number or frequency
American student
150
Asian student
30
European student
5
Mode = american student
13
When to use what
• Mean:
– No extreme scores and are not categorical
• Median
– Extreme scores and you do not want to distort the
average
• Mode
– Data are categorical in nature and values can only
fit into one class
– E.g. hair color, political affiliation, religion
14
Descriptive Statistics in Excel
• Take Figure2.9 (S-p57), input these figures to
Excel
• Data  data analysis  data analysis box 
choose Descriptive Statistics  tick “labels in
first row”  output range=c1  tick
“summary statistics”  click “OK”
15
Descriptive Statistics
Income Level
$135,456
$54,365
$37,668
$34,500
$32,456
$25,500
Income Level
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
53324.16667
16887.71824
36084
#N/A
41366.2926
1711170163
4.861219327
2.173756462
109956
25500
135456
319945
6
16
Exercise 1 (S-p62)
• Calculate mean, median and mode for the
following data:
Score1
3
7
5
4
5
6
7
8
6
5
Score2
34
54
17
26
34
25
14
24
25
23
Score3
154
167
132
145
154
145
113
156
154
123
17
Exercise 2 (S-p62)
• Writing a sale report to your boss according to
the figures of things sold today:
special
Number Sold
cost
Huge Burger
20
$2.95
Baby Burger
18
$1.49
Chicken Littles
25
$3.50
Porker Burger
19
$2.95
Yummy Burger
17
$1.99
Coney Dog
20
$1.99
18
Exercise 4 (S-p63)
• Calculate the average sale
toy
July sale
August Sale
September Sale
slammer
12345.00
14453.00
15435.00
radar zinger
31454.00
34567.00
29678.00
lazertags
3253.00
3121.00
5131.00
19
Exercise 5 (S-p63)
• Patient record
• Mean and median, which is better for what?
12/1-12/7
12/8-12/15
12/16-12/23
0-4 years
12
14
15
5-9 years
15
12
14
10-14 years
12
24
21
15-19 years
38
12
19
20